Bacon–Shor code

The Bacon–Shor code is a Subsystem error correcting code.[1] In a Subsystem code, information is encoded in a subsystem of a Hilbert space. Subsystem codes lend to simplified error correcting procedures unlike codes which encode information in the subspace of a Hilbert space.[2] This simplicity led to the first demonstration of fault tolerant circuits on a quantum computer.[3]

Given the stabilizer generators of Shor's code: , 4 stabilizers can be removed from this generator by recognizing gauge symmetries in the code to get: .[4] Error correction is now simplified because 4 stabilizers are needed to measure errors instead of 8. A gauge group can be created from the stabilizer generators:.[4] Given that the Bacon–Shor code is defined on a square lattice where the qubits are placed on the vertices; laying the qubits on a grid in a way that corresponds to the gauge group shows how only 2 qubit nearest-neighbor measurements are needed to infer the error syndromes. The simplicity of deducing the syndromes reduces the overheard for fault tolerant error correction.[5]

    ZZ   ZZ 
  q0---q1--q2
XX|    |   |XX
  |    |   |
  q6--q2--q8
XX|    |   |XX
  |    |   |
  q3--q4--q5
   ZZ   ZZ

See also

References
  1. Bacon, Dave (2006-01-30). "Operator quantum error-correcting subsystems for self-correcting quantum memories". Physical Review A. 73 (1): 012340. arXiv:quant-ph/0506023. Bibcode:2006PhRvA..73a2340B. doi:10.1103/PhysRevA.73.012340. S2CID 118968017.
  2. Aly Salah A., Klappenecker, Andreas (2008). "Subsystem Code Constructions". 2008 IEEE International Symposium on Information Theory: 369–373. arXiv:0712.4321.{{cite journal}}: CS1 maint: multiple names: authors list (link)
  3. Egan, L., Debroy, D.M., Noel, C. (2021). "Fault-tolerant control of an error-corrected qubit". Phys. Rev. Lett. Nature (598): 281–286. arXiv:2009.11482. doi:10.1038/s41586-021-03928-y.{{cite journal}}: CS1 maint: multiple names: authors list (link)
  4. Poulin, David (2005). "Stabilizer Formalism for Operator Quantum Error Correction". Phys. Rev. Lett. American Physical Society. 95 (23): 230504. arXiv:quantum-ph/0508131. doi:10.1103/PhysRevLett.95.230504.
  5. Aliferis, Panos, Cross, Andrew W. (2007). "Subsystem fault tolerance with the Bacon-Shor code". Phys. Rev. Lett. American Physical Society. 98 (22): 220502. arXiv:quant-ph/0610063. doi:10.1103/PhysRevLett.98.220502.{{cite journal}}: CS1 maint: multiple names: authors list (link)
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